Platonism CageMatch at MoMath

After spending two hours in the middle of the day hearing about unexpected uses of twistors to study particle scattering amplitudes, yesterday I went down to Manhattan’s relatively new Museum of Mathematics, which had scheduled a “Family Friday” event, featuring Edward Frenkel and Jim Holt. The event began with Frenkel giving a presentation about math, kind of an introduction to his wonderful new book Love and Math. Everyone in the audience hoped that the kids in attendance didn’t catch his comment about a typo in reference to the LHC (given Frenkel’s film experience, some had suggested that a joint event with the neighboring Museum of Sex would have been a good idea).

Things really got exciting though when Jim Holt joined him on the stage, for a no-holds-barred discussion of Platonism and mathematics in front of a standing-room-only crowd. Holt ripped into Frenkel as engaging in “mysticism” by claiming that mathematical objects are “real” and “exist”. He quoted from Bertrand Russell, who early in life took Platonist positions, but in his old age renounced them. Frenkel countered, dismissing Russell’s later quotes as those of someone who had gone soft in the head. He went on to quote arch-Platonist Kurt Gödel, with the response from Holt a low blow: he told the story of how Gödel had died a paranoid, starving himself to death. Holt continued the attack in the same vein, telling about Georg Cantor, and his end in the loony-bin. The implication was that Platonists are not just mystics but nuts.

Frenkel then decided to try taking the high road, invoking W.V.O. Quine and Hilary Putnam (distinguished non-nuts Harvard professors I took courses from) and their Indispensability Argument. The basic idea there is that the best choice of what “exists” is those entities that are an indispensable part of our best theory of the material world. Not sure yet whether twistors count, but if they become part of the new unified theory of gravity and the Standard Model, then they surely exist as much as anything does. Holt parried with Hartry Field’s Science Without Numbers: A Defence of Nominalism which supposedly shows you can do Newtonian physics without math. Frenkel (together with much of the rest of the audience) scoffed at this, making the obvious riposte: what about GR?

This was finally brought to an end with a few questions from the audience, a sizable contingent of which was underage. They seemed to be having a great time, far more entertained by this sort of thing than by the usual flashy trinkets people use to try and get them interested in math (but which seem to work better on the pre-verbal baby crowd). All in all, a highly edifying experience, I hope the Frenkel/Holt show gets taken on the road.

A logician who spent time at the Institute in the 1950s told the story of how Gödel looked up something in a book in the Institute library that he had consulted some years before. Finding that it did not say what he remembered, Gödel concluded that there are demons that go around changing things in books. This, along with his later paranoid belief that people were trying to poison him, suggests a sort of mad hyper-Platonism that strikes me as being a reasonable disqualification when it comes to the more mundane question of mathematical Platonism.

Peter,
With regard to the typo, I wasn’t there but I’ve often wondered if all sorts of trouble could have been avoided the past few years by referring to the LHC as the Baryon Collider instead? Speaking of which (slightly off topic) do you plan on reviewing Moffat’s “Cracking the Particle Code of the Universe” book containing his very detailed take on the Higgs discovery?

I agree that bald questions like “Do mathematical objects exist?” can be a waste of time, just an uninteresting argument about the meaning of words. The argument over “Platonism” that Frenkel and Holt were having I think is more interesting (besides being kind of fun), since it gets at the question of what to make of “the unreasonable effectiveness of mathematics”. Different takes on this question aren’t necessarily “right” or “wrong”, but lead one to value and take interest in different aspects of mathematics and physics. To the extent that there really are philosophical issues with substance, to me this one of how mathematics, physics and our relationship to reality fit together is a great one.

As for the great mathematicians = lunatics idea, that’s just not my experience at all. There’s nothing much to the idea that mental illness corresponds to mathematical talent and achievement. While there have been great mathematicians with mental health issues, generally they did great work while healthy, not so much while sick.

Chris Kennedy,
Probably you’re right that CERN should have gone for LBC instead of LHC to stay away from trouble…

I briefly skimmed through the Moffat book in the bookstore, don’t think I’ll carefully read it or write about it here. His take on the Higgs discovery is an unusual one, since he was prominently arguing for theories with no Higgs. Those who want to know everything about the subject may find some interesting new things in the book (and what I read looks accurate), but in general his personal skepticism about the Higgs so far just seems to have been a wrong direction, making his take on the story not of wide interest.

Peter Woit: Yes, I agree that it was a product (or aspect) of mental illness, but merely wanted to point out that a mind that does not distinguish well between imagination and reality in general is not necessarily a reliable authority on Platonism.

I also doubt that madness exists at a rate among the best mathematicians that is greater than that of top people in other creative fields, but I’m not so sure about logicians in particular. Was it not George E. P. Box who said “All model theorists are mad, but some are useful”?

You know, just because you’re paranoid, doesn’t mean you don’t have enemies — or haven’t lost some socks in the laundry.

But seriously: while a ga-ga sort of Platonism is congruent with certain kinds of mental illness, the two are not the same, and one doesn’t require the other. You’re keeping your sanity when you regard nonsensory (but not nonsensical) entities as hypothetical or contingent. You just have to avoid the ga-ga part, thinking that you’re perceiving some definitely existent nonsensory, “higher” reality through pure thought, as Descartes and Plato dreamed of.

Ask instead for some proof of relationship to sensory reality. Then you’re using your reasoning power soundly, to understand a reality that is beyond our limited senses, but somehow connected with them. Hold on to your senses, as Aristotle said, and accept form as always related to matter and matter to form.

It always interests me that most mathematicians and physicists “behave” as though they were Platonists, even when they are not. By this I mean that the language they use to describe the mathematical concepts used in their work is inherently Platonic. It is only when you press them on the issue that some of them reluctantly admit that mathematical Platonism is problematic philosophically. The point is that mathematicians and physicists “perceive” mathematical objects as though they were real, and this illusion is very compelling.

Curiously, we also “perceive” moral principles as though they were real, even though moral realism exhibits similar deficiencies as mathematical realism (unless you are willing to invoke religion). This is why I think debates about mathematical realism are important – I think that an adequate account of our perception of mathematical “truth” might shed important light on our perception of moral “truth.”

The ad hominem arguments from Jim Holt are a disgrace. If he has a convincing argument for Nominalism then he should give it; and if he doesn’t then he should lay off the personal attacks. Cantor was driven into a depression from the attacks of Kroenecker. Gödel was denied full professor status for many years due to the personal vindictiveness of one leading figure in the IAS. If both were paranoid it was also true that both had some reason to be. If personal attacks are allowed then how about we start assembling data on the marital failures of Nominalists, or their alcoholic intake, or what have you.

nasren,
I was exaggerating a bit the combative nature of the argument for effect, since I found novel and amusing the phenomenon of a lively, entertaining, high level discussion in front of a general audience of all ages of the philosophy of mathematics. Normally this sort of thing is done in a way that puts all but the most devoted to sleep. Yes, there were some low blows, but everyone knew they were low blows intended for entertainment purposes. Why do you think I described this by the term used for professional wrestling? Do you think that professional wrestlers are trying to hurt each other?

Hi Peter — yes, I understood your intention, but as a sometime contributor to this debate I can tell you that low blows are not at all uncommon. Platonism is not so much refuted as routinely abused. And this irrationalism has increased markedly in the last twenty years. The idea that Platonism is just a whacky, space-cadet view is becoming completely standard. It seems to me that post-modernism (abuse, rhetoric-as-argument, ad hominem attacks, hero worship of celebrity “thinkers”) has become the norm across all areas, including much of theoretical physics. So far only mathematics has been exempt from this corrosion. So far…

nasren,
I guess my perspective is different. Working in a math department, I don’t hear anyone arguing against Platonism, just see lots of people working happily with things that they consider very much real. Those who want to argue against Platonism, from this point of view, are not a threat but a welcome opportunity to think about the deep questions of how math, physics and reality are related.

Yes, it’s outside of the maths department that the war is being lost, in the general culture. I have to say I’m very much on Frenkel’s side in this. As long as people see maths as a mere game — as Formalists are won’t to do — it will be very much harder to bring people into maths in the future. So my concerns are not just about maths at the College level, but how it is perceived in high schools and junior schools. But my frustrations at what is happening at these early levels is beyond worries about abstract ontology.

To Nasren:
My ad hominem remarks about Gödel and Cantor were meant to be in a jocular vein. (You had to be there.) The serious argument against mathematical platonism is the “epistemic” one: if mathematical entities transcend spacetime and are causally inert, then how do we, as physically embodied beings, come to have knowledge of them? Platonists like Penrose, Gödel, Connes et al. respond to this point by waving their hands and saying consciousness “breaks through” (Penrose) to platonic reality or that we apprehend it in some extrasensory way. One practical issue here is whether (given Gödel-incompletess) we can somehow refer to a platonically unique “intended model” of Peano arithmetic–whether, that is, we have an absolute notion of finitude.
Cordially, Jim Holt

Well, Quine and Putnam may not be nuts, but then again neither is Atiyah, and he does not claim to be a Platonist (see, for example, Created or discovered?, Did we invent number theory?, and the first part of The Nature of Space). I generally find it more profitable to listen to the opinions of mathematicians themselves rather than philosophers of mathematics.

I was a Platonist as an undergraduate (it is often said that all mathematicians are when they are young) but I later realized that formalism (which, by the way, is really a term of abuse; Hilbert himself never used it, it was what his critics called him) is the more sophisticated and the better view. I agree with Jim Holt.

anon,
Thanks for the links to Atiyah, who always has wonderfully insightful things to say. Listening to these though, I’d personally describe his point of view as basically Platonist, while also taking into account some of the subtleties of the question. He describes mathematics as having a “background”, “there waiting to be discovered”. For him mathematicians do not invent mathematics, what they do is discover it in a certain way, a way that does depend on human beings and their history. His main image is of mathematical research as searching in the darkness for objects to discover. The names given to mathematical objects come from mathematicians, as does the path of the discovery. The way we think about a field of mathematics definitely depends upon its history, what was discovered and named first, what vision the discoverers had of how things fit together when they made their discoveries (as well as what parts of it remain now dark, still to be discovered). But I think the way he talks about this, the picture is primarily Platonist, one of objects out there to be discovered, while acknowledging that the order and way discoveries are made matters (he should know, having discovered more things himself than most people…).

I certainly agree that Atiyah takes a more nuanced, middle-of-the-road approach. I would personally say, however, that his position, at least as it seems to me, tends to lean toward the formalist side.

The traditional Platonist view, as I understand it, is that mathematical objects truly exist as Platonic forms and that mathematicians are somehow able to discover them. Atiyah makes the observation that the sense in which they exist is really quite trivial. What we call mathematics is simply an expression of how mathematicians, as human beings, perceive the physical world. If mathematicians did not exist, mathematics would not exist. So the way I see it, and this is how I interpret Atiyah’s position, mathematics exists as mathematical language in the brain of the mathematician. It is, of course, inspired by our experience of the physical world, but to say that mathematics exactly is the external world is a jump that is not philosophically justified. One can certainly say that mathematics represents or reflects the physical world, but only in a rather limited sense.

I think many Platonists have the impression that formalists see mathematics as a purely logical exercise in symbolic manipulation that is not connected to the physical world, but that perception is not true. Even Hilbert, of course, loved mathematical physics.

Atiyah himself makes the comparison to metaphysics, in which Kant synthesizes the views of Plato with those of Hume, who denied the existence of the external world and affirmed only our perceptions of it. In this interview, for example, Quine stubbornly insists on holding on to his materialist views, whereas it seems to the interviewer, as well as myself, that idealism is the better view (it is very probable that the powers of our faculties of perception are limited and therefore the true nature of the existence of things is forever beyond our reach – that is the Kantian prohibition, something I think the multiverse theorists should become more familiar with).

Jim — duly noted. The internet could be defined as chloroform for humour. I am guessing the tone made intentions clear.

My own view on the epistemology question (regarding mathematical Platonism) is that philosophers have screwed that question up for some eighty years or so. Epistemic access does not require causal access, and if it did a lot of non-mathematical knowledge would be in trouble as well. (Like our knowledge that we will die someday: we obviously don’t have causal signals being sent backwards from our deaths.) Philosophers have just been dumb about this.

On the Atiyah question I agree with Peter: I don’t see how one can listen to that first video “Created or Discovered” and not realise that he is espousing Platonism. The second video does not much modify it. He is and always has been a nuanced Platonist. He doesn’t believe in some super-magical ability to contact a spirit realm in which the number two exists but that idea is the nasty caricature of nominalsts — “Plato’s Heaven” said in sarcastic tones.

While manifolds may exist in nature – we are said to living in one – the open sets, the sets of coordinate patches we use to define describe them are not manifestly visible. In fact, if I understand correctly, the mathematics of manifolds is supposed to be about what independent of these methods of description. Which leads me to two questions –

1. Should it possible to do all of our familiar mathematics without the artifacts such as coordinate systems?

2. If we have to invoke things that do not exist (arbitrary constructions such as coordinate systems) in order to do mathematics, why is it so far-fetched that we have to invoke things that do not exist in nature (mathematics) in order to do physics? That is, just as coordinate patches are artifacts we need to do useful work with manifolds, which are conceived as existing independent of any particular atlas of coordinate patches; mathematics is the artifact we need to do useful work with Nature?